On the Laplace–Residual Power Series Solution of the Nonlinear time-Fractional Newell–Whitehead Equation

In this work, the Laplace–Residual Power Series Method (L-RPSM) is employed to obtain accurate analytical–numerical solutions for a class of nonlinear time-fractional evolution equations involving the Caputo fractional derivative. Such equations play an important role in applied mathematics, as they are widely used to model complex physical and biological phenomena characterized by memory and hereditary effects. The proposed methodology is first formulated for a general class of nonlinear time-fractional evolution problems, where the main steps of the L-RPSM are clearly outlined. In addition, a detailed convergence analysis is carried out to ensure the existence, uniqueness, and validity of the resulting series solutions, thereby confirming the reliability of the proposed approach from a theoretical perspective.

To demonstrate the effectiveness and practical applicability of the method, the nonlinear time-fractional Newell–Whitehead equation is considered as a test model. This equation arises naturally in the mathematical modeling of pattern formation, reaction–diffusion systems, and related processes in applied sciences. Numerical simulations are performed for different values of the fractional order, and the corresponding results are presented in tabular form as well as illustrated through two- and three-dimensional graphical representations. The influence of the fractional-order parameter on the qualitative behavior of the solutions is examined and discussed in detail. Moreover, the use of the Laplace transform within the proposed framework enhances the convergence properties and computational efficiency of the method, leading to improved accuracy of the obtained results. The results clearly indicate that the L-RPSM provides highly accurate, efficient, and reliable solutions, making it a powerful and effective tool for solving nonlinear time-fractional evolution models encountered in applied mathematics and related scientific fields.